THE IDENTITIES OF PI-RINGS
S. A. AMITSUR
1. Introduction. Let 5 be a ring with a ring of operators ß. The
ring S is said to be a Pi-ring (that is, a ring with a polynomial
identity) over ß if S satisfies a polynomial identity g(xi, • • • , xn) =0
with coefficients in ß. We make correspond to every Pi-ring 5 a universal Pi-ring which satisfies the identities and only the identities of
S. The universal ring is maximal in the sense that the rings satisfying
the identities of S are characterized as the homomorphic images of
the universal ring. The Jacobson radical [3 ] of the universal ring is
shown to be its maximal nil ideal. This fact and the study of the
universal rings which correspond to total matrix algebras yield a
representation
of the free ring over ß, which in turn implies that
every ring (not necessarily a Pi-ring) is a homomorphic image of a
subdirect sum of total matrix rings of finite order over the ring of all
integers. In particular, if the ring considered is nilpotent, the orders
of the matrix rings are bounded. Another result obtained by studying the radical of the universal rings is that every Pi-ring satisfies an
identity of the form : $£„(x) = 0 where Sin (x) is the standard polynomial
of degree 2» (see [2]).
2. Universal Pi-rings.
In what follows we assume that ß is an
integral domain which contains an infinite number of elements and
such that: (1) aS = 0, a£ß,
implies a = 0. (2) a(rs) = (ar)s=r(as)
for every a£ß and r, s E S. Let{x} be an infinite set of indeterminates
over ß. We denote by ü[x] the free ring generated by the set {x} and
ß. Following Specht [l, p. 565] we call an ideal Q in ü[x] a T-ideal
if (1) ap(x)EQ, aEtt, a = 0, implies p(x)EQ, and (2) QTQQ for
every homomorphism T: x^*t(x) of ß[;e] onto its subrings.
If 5 is a PI-ring over ß, the totality of the identities of S constitute
a nonzero P-ideal Qs in Q[x]. We refer to this ideal as the ideal of
identities of 5 and the quotient ring ß[#]/(2s will be called the
universal ring of S.
It was shown in [l] that in the case of Pi-rings of characteristic
zero, which is the case where ß is the integral domain of all integers,
the converse also holds. That is: if Q is a nonzero T-ideal, then
ß [*]/(? is a PI-ring whose ideal of identities is the ideal Q. The following lemma is a generalization of this result:
Received by the editors March 1, 1952.
27
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28
S. A. AMITSUR
[February
Lemma 1. Let P be an ideal in ßfje].1 The quotient Q[x]/P is a PIring if and only if P contains a nonzero T-ideal; and if this condition
holds, the ideal of identities of £l[x]/P is the maximal T-ideal contained
in P.
Proof. The set of all polynomials g(x)EP such that g(x)TEP for
every homomorphism T constitute the maximal P-ideal Q contained
in P. If Qt^O, let g(xi, ••-,*»)
be any nonzero polynomial of Q.
Since Q is a T-ideal, for any set of polynomials h(x), • • • , tn(x),
g(h(x), • • • , t„(x))EQ- Thus if ' denotes reduction modulo P, it fol-
lows by P2<3 that:
g'(h(X), •■■ , tn(x)) = g(ti(x),
■■■ , *.'(*)) = 0.
This proves that g(xu ■ ■ • ,x„)=0
is an identity
satisfied by
ß[x]/P.
Conversely,
if g(xi, ■ ■ • , xn) =0 is satisfied by ß[x]/P,
for any set of polynomials h(x), • • • , tn(x) we have g(t{(x), • • • ,
t„ (x)) =0. This means that g(h(x), ■ • • , tn(x)) EP, which proves that
the T-ideal generated by g(x) is a P-ideal contained in Q. Hence
g(x)EQ and the proof is completed.
Note that if the set {x} is sufficiently large, the ring 5 is a homomorphic image of its universal ring. Let R be a PI-ring which satisfies
the identities of S; then Qr^Qs, where Qr and Qs are the ideals of
identities of R and S respectively. Hence the universal ring of R is
a homomorphic image of the universal ring of S. In particular, if the
cardinal number of the set \x\ is not smaller than the cardinal numbers of R and S, the ring R is also a homomorphic image of the universal ring of S. On the other hand, by Lemma 1 the universal ring of
5 satisfies the identities of S; it follows, therefore, that every homomorphic image of the universal ring of 5 satisfies the identities of S.
Thus we have proved:
Theorem 1. A ring R satisfies all identities of S if and only if R is a
homomorphic image of the universal ring of S.
It should be remarked that if the set {x} is fixed, the preceding
theorem holds only for rings R with cardinal number not greater than
that of the set {x} ; but if R is given, one can choose {x} so that the
preceding theorem will be true.
In what follows "radical" and "semisimplicity" will be used in the
sense of Jacobson [3].
Let Q be a nonzero JMdeal.
Lemma 2. Let p(xu • ■ ■, x„)EQ and let
1 We assume that from ap^O, a£ a, ap(x)EP
it follows that p(x)EP-
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I953I
THE IDENTITIESOF PI-RINGS
P(xi, ■••,*»)«"£
29
p,(xx, ■ ■ • , Xn)
where each p, is homogeneous in Xi and of degree v in Xx\ then p,EQ,
v = 0, 1, • • • , m.
The proof of this lemma is similar to that of [4, Lemma 3]. That is:
let atGß, then p(a{xi, • ■ • , x„) = 2» a'iPÁxi, • • • • xn). Choosing m
different elements «i, • ■ • , am oí ß, one can find also X,„EQ such that
2i hi,p(aiXi, • • • , xn) =Ap,(xi, • • • , xn), where A= H«* («»—«*)•
This implies Ap,(x)EQ; hence p,(x)EQLemma 3. Lei p =p(xi, ■ • • , xn) be a homogeneous polynomial in X\.
The polynomial p is quasi regular modulo Q if and only if p is nilpotent
modulo Q.
It is obvious that if p is nilpotent mod Q, then p is also quasi regular
mod Q. Conversely, if p is quasi regular, then p —q+pqEQ
for some
polynomial q. Hence q=.p+pq=.p+p2+pq=.
• • ■ ==p+p*+ . . .
+pn+l+pn+lq
(mod Q). Let q= ^7-o <Z*De the decomposition
of q as
a sum of homogeneous polynomials in Xi of degrees v = 0, 1, • ■ • , m.
Thus g= £î+1 p»+ Er.o Pn+%~ ET-0 Î.G0- Now choose »>».
Then one observes that one of the elements of the decomposition of
g as a sum of homogeneous polynomials in Xi of different degrees must
be p"; hence by the preceding lemma it follows that p"EQ- q.e.d.
Theorem 2. // 5 is a PI-ring without nilpotent ideals, then the universal ring R' = ß [*]/(?s is semisimple.
Proof. Let p(xu • • • , xn) be a polynomial belonging to the radical2
of R'. Take as xn+i any of the indeterminates of the set {x} not appearing in p(x). Evidently, p(x)xn+i also belongs to the radical. Since
this polynomial is homogeneous in xn+i, the preceding lemma yields
(p(x)xn+x)mEQ for some integer m. This evidently implies that if
Sx, • ■ ■ , sn are any elements of S, the ideal p(si, • • • , sn)S is a nil
ideal. From the results of [5] it follows that Pi-rings which do not
contain nilpotent ideals also do not contain right nil ideals; hence
P(si, • • • , sn)S = 0. The absence of nilpotent ideals implies also that
P(si, ■ • • , sn) =0. Thus S satisfies the identity p(xi, • ■ • , xn) =0.
By definition of Qs, p(x)EQs and the proof is completed.
Applying the preceding result we are able to prove :
Theorem 3. // Q is a nonzero T-ideal, then the radical of ß| x\ /Q is
its maximal nil ideal.
1 This means, of course, "p is a representative
of a class of the quotient Si[x]/Qs
which belongs to the radical."
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30
S. A. AMITSUR
In fact,
we show that
the
radical
[February
coincides
with
the
lower
radical [6]. Let J, L be the ideals of fl[x]. such that J/Q, L/Q are
respectively the radical and the lower radical of ß [#]/(). It is
well known that J'Q.L'DQ. The quotient ß[x]/L is a PI-ring3 since
it is a homomorphic image of ß [*]/(?> hence by Lemma 1, L contains
the ideal of identities L0 of this ring. Since Q,[x]/L^(Q,[x]/Q)/(L/Q),
£l[x]/L does not contain nilpotent
the preceding theorem that ß[#]/Lo
=(J, Lo)/Lo. Since the first is an
regular ring J/Q, it is also quasi
(J, Lo)/Lo is quasi regular which is
ideals; it follows, therefore, by
is semisimple. Now J/(J(~\Lo)
isomorphic image of the quasi
regular. But this implies that
possible only if (J, L0) =Lo, i.e.,
L0^2J. Thus /2L2Lo3/,
which completes the proof.
The equality J=Lo proves that:
Corollary
1. The ideal J is a T-ideal.
3. The free ring Q[x]. There is particular
interest
in the ideals
of identities of the total matrix rings ßn of order « over ß. Let Mn
denote the ideal of identities of ßn, then we have ilfi^Mj^
■■
Each of the quotients ß [x]/Af„ is, by Theorem 2, a semi simple PIring. From [7, Theorem 2] we deduce that these rings are subdirect
sums of central simple algebras of bounded degree. In the present
case we can obtain the stronger result:
Lemma 4. The quotient Sl[x]/Mn is a subdirect sum of total matrix
rings of order n over ß.
Indeed, let r be a homomorphism:
Xr~*fi of ß[x] onto ßn, where
{r<} is a set of generators of ß„. Let QT be the kernel of t. Thus
Q[x]/QT=Çln. If r ranges over all possible homomorphisms
of this
type, then (]QT= Mn. For, by Lemma 1, it follows that4 QT=\Mn for
every
r. Hence
(iQr^Mn.
Conversely,
if g(xit • • • , xn)EÇ\Qr,
let
»"i, • • • , r„ be any set of matrices of ß„. Let r be the homomorphic
mapping: #<—»r,-,i=l, • • • , n, and let the correspondence x—*r for
the rest of the set {x \ be defined so that the set {r} will contain a
set of generators of ß„. Then we obtain g(n, ■ ■ ■ , rn) =0. This proves
that g(xi, • • • , xn)EMn, and the proof is completed.
A subdirect sum of rings isomorphic with ß„ is a subring of a
complete direct sum of such rings. The latter is isomorphic with a
total matrix ring of order « over a complete direct sum R of rings isomorphic with ß. The ring R is known to be free of nilpotent elements
since ß is an integral domain. Hence :
* One readily verifies that L satisfies the assumption of the footnote
4 QT evidently satisfies the assumption of footnote 1.
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1.
19531
THE IDENTITIES OF PI-RINGS
31
Corollary
2. ß[x]/Jlf„ is isomorphic with a subring of a total
matrix ring of order « over a commutative ring which does not contain
nilpotent elements.
It follows, therefore, by Theorem 1 that:
Theorem 4. A ring S satisfies all identities of the total matrix ring ß„
if and only if it is isomorphic with a subdirect sum of rings isomorphic
with ß„; an alternative necessary and sufficient condition is that S be
isomorphic with a subring of a total matrix ring of order n over a commutative ring.
Lemma 5. f\Mn=0.
For if g(xi, • ■ • , xm)EMn, then g = 0 is satisfied by the ring of all
finite matrices over ß but no such identity exists by consequence 2 of
[5]. Hence g = 0, i.e., fl-^» = 0It follows now in view of Lemma 4 that:
Corollary
3. The free algebra ß [x] is a subdirect sum of total matrix
rings of finite order over ß.
We now apply the preceding result to the case ß = 7, the ring of
all integers. Since every ring is a homomorphic image of the free ring5
l[x], we obtain:
Theorem 4. Every ring is a homomorphic image of a subdirect sum
of total matrix rings of finite order over the ring of all integers.
Since the ring / is a subdirect
theorem yields:
sum of prime fields, the preceding
Corollary
4. Every ring is a homomorphic image of a subdirect sum
of total matrix rings of finite order over prime fields.
Consider the T-ideal Nn oí l[x] generated
by the polynomial
XiX2 • • • xn. This ideal contains all the polynomials with degree à».
From consequence 2 of [S] we know that the minimal degree of the
polynomials of the ideal Mk is greater than or equal to 2k. Hence
Nn^Mk where k = [(« + l)/2]. Since every nilpotent ring is a homomorphic image of l[x]/Nn,6 we obtain, by Theorem 3:
Theorem 5. Every nilpotent ring of index of nilpotency « is a homomorphic image of a subdirect sum of total matrix rings over I of order
bounded by [(«+l)/2].
By the arguments
of Theorem
3 this readily implies:
8 If the set {x} is sufficiently large.
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32
S. A. AMITSUR
Corollary
[February
5. Every nilpotent ring of index of nilpotency « is homo-
morphic with a subring of a total matrix ring of order [(« + l)/2]
commutative ring which does not contain nilpotent elements.
over a
Remark. If one considers algebras over a field F, then Pean replace
the integral domain / of all integers, and in this case the commutative
rings obtained in Theorems 4, 5 and in Corollary 5 are either F or
commutative
algebras over F.
4. The semisimple
present
universal
Pi-rings.
The main object of the
section is to prove:
Theorem 6. The ideals Mn are the only T-ideals P of ß[x] such that
ß[x]/P is semisimple.6
To prove this theorem we need the following generalization
Lemma 3]:
of [4,
Lemma 6. Let C be a commutative ring with a unit element. The ideals
of identities of a PI-ring S and of the direct product SXC (over any extension of ß) coincide.
The proof of this lemma was published in Hebrew in [8]. The following proof is simpler.
Put Xj= ^2i un, and write g( ^uu,
• • • , ^m«) = IZ* g*(«y») as a
sum of homogeneous polynomials gk- It follows now by Lemma 2
that the identities gk= 0 are satisfied by 5. Let x¡ = a¡, a¡ESXC,
then a¡= 2Z¿ SjíCjí,SjíES, c¡íEC. Since the polynomials gk are homogeneous, it follows that gk(sjiCji)=gk(sji)c = 0, where c is an element of C. By substituting
uh = SjíC¡í we immediately
obtain that
g(au ■ ■ • , an)=0. q.e.d.
The ideal Mn is evidently also the ideal of identities of the total
matrix ring Rn of order « over any commutative
ring R which possesses a unit element. Now if A is any central simple algebra of order
«2 over its center F, one can find an extension F oî F such that
A X F=Fn. In view of these facts, the preceding lemma implies that
the ideal of identities of any central simple algebra A of order n2
over its center is also Mn. With the aid of this result we can now prove
Theorem 6.
Let P be a P-ideal such that ß[x]/P is semisimple. Let k be the
minimal degree of the polynomials of P. By [7, Theorem 2] it follows
that this ring is a subdirect sum of central simple algebras Am such
6 This is equivalent to the fact that the only universal semisimple Pl-rings are
the quotients íí[x]/ilí„.
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19531
THE IDENTITIES OF PI-RINGS
33
that «2 is the upper bound of their orders over their centers. Since
the ideal of identities of these algebras are the ideals Mi, it follows
readily that the ideal of identities of the subdirect sum is Mn. Thus,
Lemma 1 yields that P = Mn, q.e.d.
Let 5 be a PI-ring. By Theorem 6 and by Corollary 1 it follows that
the radical
of the universal
ring
ß[x]/Cs
is the quotient
ideal
Mn/Qs-1 From the main result of [2] it follows that Mn, «^1, contains the standard
polynomial
S2n(x) = X)±xh ' * " *•», where the
sum ranges over all permutations
(i) of 2» letters and where the sign
is taken positive for even permutations and negative for odd permutations. We shall use also the notation: So(x) =Xi. Applying Theorem
3 we now obtain that S2n(x)mEQa for some integer m. Hence:
Corollary
6. Every PI-ring
satisfies the identity S2n(x)m = 0.
The preceding corollary was proved under the assumption that ß
is infinite; nevertheless, it is true for every PI-ring. For if ß is a
finite integral domain, we consider the ring S[t] of all polynomials
over 5 in a commutative
indeterminate
/. Since 5 satisfies linear
identities [4, Lemma 2], S[t] also satisfies these identities. This ring
can be considered as a PI-ring over the infinite integral domain
ß[/]. Hence, by the preceding corollary, S[t], and therefore also S,
satisfy the identity S2n(x)m = 0.
Denote by S„ the P-ideal generated by the standard polynomial
S2n(xi, ■ • ■ , Xtn). That is, 2„ contains the polynomial of the form
(I)
h(x) = £ a(x)S2n(fi(x), ■■■ , f2n(x))b(x).
From the main theorem of [2] we readily deduce that the total
matrix ring ß„ satisfies the identities A(x)=0, A(x)£2„.
Naturally,
this suggests the question whether the polynomials of 2„ are the
only polynomial
identities satisfied by ßn. In the notation of the
present paper this problem is equivalent
to the assertion of the
equality Af„=2„.
From [2] it follows that Afn32„. On the other hand, consequence
3 of [5] implies that the minimal degree of the polynomials of Mn+i
is greater than or equal to 2(« + l). Thus, the results of Corollary 1,
Theorems 6 and 3 imply that Ain/Sn is the maximal nil ideal of
ß[x]/Sn. This yields the following characterization
of the polynomial
identities satisfied by ßn (i.e., the polynomials which belong to Mn):
Theorem 7. The identity g(x) =0 is satisfied by ß„ if and only if for
We use the notation Afo= ß[*].
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34
S. A. AMITSUR
some integer m and for some indeterminate
the polynomial (g(x)xk)m has the form (I).
Xt not appearing
in g(x),
This theorem provides only a partial answer to the problem of determining the identities satisfied by ß„. We conclude with the remark
that the preceding theorem reduces the proof of a positive answer to
the question raised above to the assertion that the quotient ring
ß[x]/2„ does not contain nilpotent elements. This is the case for
» = 1, since ß[x]/2i is a commutative
ring without zero divisors,
hence il/i=2i. We conjecture that this holds for every «.
Bibliography
1. W. Specht, Gesetzein RingenI, Math. Zeit. vol. 52 (1950)pp. 557-589.
2. S. Amitsur and J. Levitzki, Minimal identities for algebras, Proceedings of the
American Mathematical Society vol. 1 (1950) pp. 449-463.
3. N. Jacobson,
The radical and semisimplicity for arbitrary rings. Amer. J. Math,
vol. 67 (1945)pp. 300-320.
4. I. Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. vol. 54
(1948)pp. 575-580.
5. J. Levitzki, A theorem on polynomial identities, Proceedings of the American
Mathematical Society vol. 1 (1950) pp. 331-334.
6. R. Baer, Radicalideals,Amer. J. Math. vol. 65 (1943)pp. 537-568.
7. S. A. Amitsur, An embedding of Pi-rings,
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8. -,
On a lemma of Kaplansky, Riveon Lematematica
48. (In Hebrew with an English summary.)
Hebrew
University
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vol. 3 (1948) pp. 47-